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Math Help - Find the supremum and infimum of the following set

  1. #1
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    Find the supremum and infimum of the following set

    Let A = {n^2/2^n | n in N}. Find the supremum and infimum, proving your assertions.

    Attempt at Solution

    SupA:

    The highest term in the set is 9/8 and thus it is the supremum, by definition.

    InfA:

    The terms in the set approach 0 as n becomes arbitrarily large. It is clear that 0 < n^2/2^n for all n, thus, 0 is a lower bound.

    I'm not sure how to prove that 0 is the infimum. Can someone lead me in the right direction?
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  2. #2
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    Re: Find the supremum and infimum of the following set

    Quote Originally Posted by My Little Pony View Post
    Let A = {n^2/2^n | n in N}. Find the supremum and infimum, proving your assertions.
    InfA:
    The terms in the set approach 0 as n becomes arbitrarily large. It is clear that 0 < n^2/2^n for all n, thus, 0 is a lower bound.
    I'm not sure how to prove that 0 is the infimum. Can someone lead me in the right direction?
    You have already shown that 0 is a lower bound for the set.
    Now show if c>0 then there is a k\in\mathbb{N} such that 0<\frac{k^2}{2^k}<c.
    Thus proving that no number greater that 0 is a lower bound.
    That can be done by using the fact that \left(\frac{n^2}{2^n}\right)\to 0.
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